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Abhishek K. Shrivastava October 2 nd , 2009

Listing Unique Fractional Factorial Designs – II. Abhishek K. Shrivastava October 2 nd , 2009. Outline. Fractional Factorial Designs (FFD). RECAP Sequential generation of design catalogs Design isomorphism problem Designs as graphs. Listing Unique designs. Graphs & designs. FFDI & GI.

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Abhishek K. Shrivastava October 2 nd , 2009

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  1. Listing Unique Fractional Factorial Designs – II Abhishek K. Shrivastava October 2nd, 2009

  2. Outline • Fractional Factorial Designs (FFD) RECAP • Sequential generation of design catalogs • Design isomorphism problem • Designs as graphs • Listing Unique designs • Graphs & designs • FFDI & GI Solving GI – canonical labeling (nauty)Implications to generating design catalogs Abhishek K. Shrivastava, TAMU

  3. add column/ factor add column/ factor add column/ factor add column/ factor … 24 Full factorial 5-factor FFD 6-factor FFD 7-factor FFD Sequential generation of FFDs • Consider 16-run designs – sequential generation • How do you pick these columns?? FFD class • Regular FFD: defining relation E=AB, F=AC, G=BD • Orthogonal arrays: added column keeps orthogonal array property • All possible choices of columns gives the catalog Abhishek K. Shrivastava, TAMU

  4. discard isomorphs 7-factor designs from 6-factor designs ... 24 design Non-isomorphic 5-factor designs Non-isomorphic 6-factor designs Non-isomorphic 7-factor designs Intermediate step Sequential generation of FFD catalogs • Consider sequential generation of 16-run designs • Note: reducing # intermediate designs will speed up the algorithm • How to discard isomorphs? Abhishek K. Shrivastava, TAMU

  5. FFD Isomorphism (FFDI) • Definition. Two FFD matrices are isomorphicto each other if one can be obtained from the other by • some relabeling of the factor labels, level labels of factors and row labels. B↔C E↔F Abhishek K. Shrivastava, TAMU

  6. … … Proposed FFDI solution (in a nutshell) • Graph models (bipartite, vertex-colored) for FFDs (2/multi/mixed-level, regular/non-regular, split-plot) • Equivalence between FFDI and GI • Solving GI Construct graphs from FFDs Solve graph isomorphism problem Abhishek K. Shrivastava, TAMU

  7. GI and FFDI • Solving GI: canonical labeling • Implications to listing FFDs efficiently Abhishek K. Shrivastava, TAMU

  8. f(A) = H f(B) = G f(C) = F f(D) = E A D E H B C F G Graph Isomorphism Problem • Direct comparison of two graphs – search for an f between G1 & G2 • Good if many isomorphisms between two graphs • Canonical labeling – Compute a signature C() defined s.t. C(G1)=C(G2) iff G1 & G2 isomorphic • Good if many graphs to compare GI Problem. Given two graphs G1, G2does there exist a bijective function f:V(G1) V(G2) that preserves vertex adjacencies? Abhishek K. Shrivastava, TAMU

  9. Canonical labeling in nauty(McKay 1981) nauty Input • graph G Output • Canonically labeled graph C(G) • Automorphisms Abhishek K. Shrivastava, TAMU

  10. Partitionwith 3 cells Canonical labeling by example • degreed(v,U), vV, UV • # edges between v and U Abhishek K. Shrivastava, TAMU

  11. No further refinement of partition possible using ‘degree’ • Exchanging labels between vertices in different cells gives non-isomorphic graphs • Try relabeling AE({A,F} is an edge, {E,F} is not!) • What about AB (same cell)?

  12. 1 2 3 4 5 6 1 2 3 4 5 6 Canonical labeling by example • Using the {C,F} split may result in a different candidate for canonically labeled graph • C(G) is one among these alternatives Canonically labeled graph C(G) with {E,A,B,C,F,D}(or {E,B,A,F,C,D}) {1,2,3,4,5,6} Abhishek K. Shrivastava, TAMU

  13. 1 2 3 4 5 6 1 2 3 4 5 6 Finding C(G) among alternatives • Pick the smallest ‘candidate’ based on some total ordering • E.g., a binary number b(G) • nauty uses this and some other rules to quickly find the canonical graph Concatenate columns [000001, 001010, 010100, 001011, 010101, 100110] Concatenate rows b(G)=0000010010100101000010110101011001102 Abhishek K. Shrivastava, TAMU

  14. Split non-singleton (search tree) Use degree to refine partitions 0000010010100101000010110101011001102 0000010010100101000010110101011001102 Summary of Canonical labeling algorithm Use degree to form partitions Vertex invariant Many vertex invariants exist Find automorphisms and C(G) from discrete partitions

  15. discard isomorphs 7-factor designs from 6-factor designs ... 24 design Non-isomorphic 5-factor designs Non-isomorphic 6-factor designs Non-isomorphic 7-factor designs Intermediate step Sequential generation of FFD catalogs • Sequential generation of 16-run designs • We know how to discard isomorphs • Note: reducing # intermediate designs will speed up the algorithm Abhishek K. Shrivastava, TAMU

  16. Implications of Graph approach to Seq. Gen. • Note: reducing # intermediate designs will speed up the algorithm • Canonical labeling • # expensive computations for comparing m designs = m • Using automorphisms to reduce # intermediate designs Abhishek K. Shrivastava, TAMU

  17. Automorphisms & Intermediate designs • Example: n=6, S = {I, ABE, ACF, BCEF} • 6-factor 2-level regular fractional factorial design • BC, EF is an automorphism Abhishek K. Shrivastava, TAMU

  18. Example contd. • 7-factor designs from the 6-factor design • Add defining words ADG or BDGor CDGorABCG, etc… • Consider graphs obtained by using defining words • BDG S1 = {I, ABE, ACF, BDG, BCEF, ADEG, ABCDFG, CDEFG} • CDGS2 = {I, ABE, ACF, CDG, BCEF, ADFG, ABCDEG, BDEFG} d1 d2 Abhishek K. Shrivastava, TAMU

  19. Example contd. + = BC, EF (isomorphism) BC, EF (isomorphism) BC, EF (automorphism) + = Abhishek K. Shrivastava, TAMU

  20. ... Non-isomorphic 2n–pdesigns a 2n–p design d Find automorphisms of d repeat for each d Find non-isomorphic defining words for d Construct 2(n+1) – (p+1) designs by adding defining words to d Intermediate 2(n+1) – (p+1) designs discard isomorphs by Graph-based check Non-isomorphic 2(n+1) – (p+1)designs ... Reducing Intermediate designs with Autom. Abhishek K. Shrivastava, TAMU

  21. Results: Computational efficiency We compare three methods for 2-level regular FFDs: • EigVal – Lin & Sitter (2008)’s algorithm • GBAnoR – Same as EigVal, except using our new isomorphism check • GBA – GBAnoR with our design reduction method added Abhishek K. Shrivastava, TAMU

  22. Results: Cumulative CPU times (in secs.) • GBAnoR vs. EigVal – reduction over 90%in most cases • GBA vs. EigVal – reduction over 97%in most cases Abhishek K. Shrivastava, TAMU

  23. Results: Number of intermediate designs • EigVal & GBAnoR give same # designs (no additional reduction) • GBA further reduces intermediate designs by 30–70% in most cases Abhishek K. Shrivastava, TAMU

  24. Results: design catalogs • Generated new catalogs of 1024 (R ≥ 6), all 2048-run (R ≥ 7) & 4096-run (R ≥ 8) designs [1967] Draper & Mitchell [1993] Chen & Wu (64-run) [2008] Lin & Sitter (512-run) [2009] Shrivastava & Ding (4096-run) Abhishek K. Shrivastava, TAMU

  25. Results for 2-level regular split-plot FFDs • Catalogs of non-isomorphic minimum aberration FFSPs New catalogs *Up to 10 factor 32-run designs appear in Bingham and Sitter (2001) Abhishek K. Shrivastava, TAMU

  26. Summary & Contributions • Generic framework for generating catalogs of non-isomorphic FFDs • New, efficient isomorphism check • Fast design generation algorithm • Extensible to different classes of FFDs by constructing graph representations • New catalogs of designs up to 4096 runs, much more than existing in current literature Abhishek K. Shrivastava, TAMU

  27. 1 2 3 4 5 6 7 8 EQ6 Fixtures EQn 1 2 3 4 5 6 7 8 9 … 1 1 … … Buses EQ2 EQ1 EQ5 Equipments EQ3 1 1 1 EQ4 … 1 2 … 1 … … … … Schematic of phone quality testing system A Related Problem: Complicated Engg. Designs Colored graph representation of a test configuration Abhishek K. Shrivastava, TAMU

  28. Thank you! Abhishek K. Shrivastava, TAMU

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